Study On Variational Bounds For Piezoelectric Composites With Imperfect Interfaces

The prediction of the overall properties of piezoelectric composites has become a long standing issue in micromechanics since such materials play an important role in the domain of smart materials. In most practical situations, the interfaces between the constituent phases of piezoelectric composites are imperfect. In the context of piezoelectricity, two special interface models and one general interface have been mainly proposed and developed, i.e., piezoelectric spring-layer and coherent imperfect interface models, piezoelectric general imperfect interface model. The piezoelectric spring-layer interface model stipulates that, across an interface, the displacement and electric potential are discontinuous while the traction and normal electric displacement are continuous and proportional to the displacement and electric potential jumps, respectively. According to piezoelectric coherent imperfect interfaces, the displacement and the electrical potential are continuous while the traction and the normal electrical displacement are discontinuous across an interface and proportional to the surface gradient of the displacement and the electrical potential, respectively, by the so-called Laplace-Young equation. The general piezoelectric interface model was derived by exploiting the idea of replacing an interphase by an imperfect interface and by an asymptotic analysis. In this model, the displacement vector, the electric field, the traction vector and the normal electric displacement field are discontinuous across an interface. The present work aims to establish the bounds for the effective properties of piezoelectric composites with two special imperfect interfaces.To achieve this objective, firstly, piezoelectric spring-layer and coherent imperfect interface models are deduced by the works of Gu and He. Secondly, the classical minimum potential principles of linear piezoelectricity are extended to such inhomogeneous material and to formally setting bounds for their effective piezoelectric properties. Thirdly, we consider a transversely isotropic piezoelectric composite consisting of a matrix reinforced by cylindrical inhomogeneities via linearly piezoelectric coherent imperfect interface which is subjected to the uniform anti-plane mechanical load and in-plane electrical load boundary conditions. By taking simple trial strain and electrical displacement or simple trial traction and electrical potential couple fields, the first-order upper and lower bounds are explicitly derived for the corresponding elastic, piezoelectric and dielectric moduli of such composite by using the established variational principles. Finally, numerical results of the obtained bounds are provided to illustrate their size-dependence.